| Variable | Description | Type |
|---|---|---|
| diabetes_dx | Diabetes diagnosis (1 = Yes, 0 = No) based on medical questionnaire. | Categorical |
| age | Age of participant in years. | Continuous |
| bmi | Body Mass Index (BMI) in kilograms per square meter (kg/m²), calculated from measured height and weight. | Continuous |
| sex | Sex of participant (Male or Female). | Categorical |
| race | Race/Ethnicity (e.g., Non-Hispanic White, Non-Hispanic Black, Mexican American, etc.). | Categorical |
| WTMEC2YR | Examination sample weight for MEC (Mobile Examination Center) participants. | Weight |
| SDMVPSU | Primary Sampling Unit (PSU) used for variance estimation in complex survey design. | Design |
| SDMVSTRA | Stratum variable used to define strata for complex survey design. | Design |
| age_c | Age variable centered and standardized (z-score). | Continuous |
| bmi_c | BMI variable centered and standardized (z-score). | Continuous |
| wt_norm | Normalized survey weight (WTMEC2YR divided by its mean, for model weighting). | Weight |
Bayesian Logistic Regression - Application in Probability Prediction of disease (Diabetes)
CapStone Project_2025
Slides: slides.html ( Go to slides.qmd to edit)
Introduction
Diabetes mellitus (DM) is a major public health concern closely associated with factors such as obesity, age, race, and gender. Identifying these associated risk factors is essential for targeted interventions D’Angelo and Ran (2025). Logistic Regression (traditional) that estimates the association between risk factors and outcomes is insufficient in analyzing the complex healthcare data (DNA sequences, imaging, patient-reported outcomes, electronic health records (EHRs), longitudinal health measurements, diagnoses, and treatments. Zeger et al. (2020). Classical maximum likelihood estimation (MLE) yields unstable results in samples that are small, have missing data, or presents quasi- and complete separation.
Bayesian hierarchical models using Markov Chain Monte Carlo (MCMC) allow analysis of multivariate longitudinal healthcare data with repeated measures within individuals and individuals nested in a population. By integrating prior knowledge and including exogenous (e.g., age, clinical history) and endogenous (e.g., current treatment) covariates, Bayesian models provide posterior distributions and risk predictions for conditions such as pneumonia, prostate cancer, and mental disorders. Parametric assumptions remain a limitation of these models.
In Bayesian inference Chatzimichail and Hatjimihail (2023), Bayesian inference has shown that parametric models (with fixed parameters) often underperform compared to nonparametric models, which do not assume a prior distribution. Posterior probabilities from Bayesian approaches improve disease classification and better capture heterogeneity in skewed, bimodal, or multimodal data distributions. Bayesian nonparametric models are flexible and robust, integrating multiple diagnostic tests and priors to enhance accuracy and precision, though reliance on prior information and restricted access to resources can limit applicability. Combining Bayesian methods with other statistical or computational approaches helps address systemic biases, incomplete data, and non-representative datasets.
The Bayesian framework described by R. van de Schoot et al. (2021) highlights the role of priors, data modeling, inference, posterior sampling, variational inference, and variable selection.Proper variable selection mitigates multicollinearity, overfitting, and limited sampling, improving predictive performance. Priors can be informative, weakly informative, or diffuse, and can be elicited from expert opinion, generic knowledge, or data-driven methods. Sensitivity analysis evaluates the alignment of priors with likelihoods, while MCMC simulations (e.g., brms, blavaan in R) empirically estimate posterior distributions. Spatial and temporal Bayesian models have applications in large-scale cancer genomics, identifying molecular interactions, mutational signatures, patient stratification, and cancer evolution, though temporal autocorrelation and subjective prior elicitation can be limiting.
Bayesian normal linear regression has been applied in metrology for instrument calibration using conjugate Normal–Inverse-Gamma priors Klauenberg et al. (2015). Hierarchical priors add flexibility by modeling uncertainty across multiple levels, improving robustness and interpretability. Bayesian hierarchical/meta-analytic linear regression incorporates both exchangeable and unexchangeable prior information, addressing multiple testing challenges, small sample sizes, and complex relationships among regression parameters across studies Leeuw and Klugkist (2012)
A sequential clinical reasoning model Liu et al. (2013) Sequential clinical reasoning models demonstrate screening by adding predictors stepwise: (1) demographics, (2) metabolic components, and (3) conventional risk factors, incorporating priors and mimicking clinical evaluation. This approach captures ecological heterogeneity and improves baseline risk estimation, though interactions between predictors and external cross-validation remain limitations.
Bayesian multiple imputation with logistic regression addresses missing data in clinical research Austin et al. (2021) in clinical research by classifying missing values (e.g., patient refusal, loss to follow-up, mechanical errors) as MAR, MNAR, or MCAR. Multiple imputation generates plausible values across datasets and pools results for reliable classification of patient health status and mortality.
Aims
The present study aims performs Bayesian logistic regression to predict diabetes status and evaluate the associations between diabetes and predictors (body mass index (BMI), age (≥20 years), gender, and race). The study anakyzes a retrospective dataset (2013–2014 NHANES survey data). It is based on a complex sampling design, characterized by stratification, clustering, and oversampling of specific population subgroups, rather than uniform random sampling. A Bayesian analytical approach addresses challenges posed by dataset anomalies such as missing data, complete case analysis, and separation that limit the efficiency and reliability of traditional logistic regression in predicting health outcomes.
Method
Bayesian Logistic Regression
The study employs Bayesian logistic regression to estimate associations between predictors and outcome probabilities.
The Bayesian framework integrates prior information with observed data to generate posterior distributions, allowing direct probabilistic interpretation of parameters.
This approach provides flexibility in model specification, accounts for parameter uncertainty, and produces credible intervals that fully reflect uncertainty in the estimates.
Unlike traditional frequentist methods, the Bayesian method enables inference through posterior probabilities, supporting more nuanced decision-making and interpretation.
Model Structure
Bayesian logistic regression is a probabilistic modeling framework used to estimate the relationship between one or more predictors (continuous or categorical) and a binary outcome (e.g., presence/absence of disease).
It extends classical logistic regression by combining it with Bayesian inference, treating model parameters as random variables with probability distributions rather than fixed point estimates.
The logistic model relates the probability of an outcome ( Y = 1 ) to a linear combination of predictors through the logit link function:
[ (P(Y = 1)) = _0 + _1 X_1 + _2 X_2 + + _k X_k ]
logit(pi)=β0+j=1∑pβjxij
p_i: the probability of the event (e.g., having diabetes) for individual i. “logit”(p_i)=log(p_i/(1-p_i )): the log-odds of the event. β_0: the intercept — the log-odds of the event when all predictors x_ij=0. β_j: the coefficient for predictor x_j, representing the change in log-odds for a one-unit increase in x_ij, holding other variables constant. ∑_(j=1)^p β_j x_ij: the combined linear effect of all predictors.
In the Bayesian framework, the coefficients ( ) are assigned prior distributions, which are updated in light of the observed data to yield posterior distributions.
Bayesian Approach
The Bayesian approach naturally incorporates uncertainty in all model parameters.
It combines prior beliefs with observed data to produce posterior distributions according to Bayes’ theorem:
[ ]
Likelihood: Represents the probability of the observed data given the model parameters (as in classical logistic regression).
Prior: Encodes prior knowledge or beliefs about parameter values before observing the data.
Posterior: Represents updated beliefs about parameters after observing the data.
Prior Specification
A weakly informative Student’s t-distribution prior, student_t(3, 0, 10), was used for regression coefficients (van de Schoot et al., 2013).
This prior:
- Has 3 degrees of freedom (( = 3 )), producing heavy tails that allow for occasional large effects.
- Is centered at 0 (( = 0 )), reflecting no initial bias toward positive or negative associations.
- Has a scale parameter of 10 (( = 10 )), allowing broad variation in possible coefficient values.
Such priors improve stability in models with small sample sizes, high collinearity, or potential outliers.
Advantages of Bayesian Logistic Regression
- Uncertainty quantification: Produces full posterior distributions instead of single-point estimates.
- Credible intervals: Provide the range within which a parameter lies with a specified probability (e.g., 95%).
- Flexible priors: Allow incorporation of expert knowledge or results from previous studies.
- Probabilistic predictions: Posterior predictive distributions yield direct probabilities for future observations.
- Comprehensive model checking: Posterior predictive checks (PPCs) evaluate how well simulated outcomes reproduce observed data.
Posterior Predictions
Posterior distributions of the coefficients are used to estimate the probability of the outcome for given predictor values.
This enables statements such as:
> “Given the predictors, the probability of the outcome lies between X% and Y%.”
Posterior predictions incorporate two sources of uncertainty:
- Parameter uncertainty: Variability in estimated model coefficients.
- Predictive uncertainty: Variability in future outcomes given those parameters.
Model Evaluation and Diagnostics
Model quality and convergence were assessed using standard Bayesian diagnostics:
- Convergence diagnostics: Markov Chain Monte Carlo (MCMC) performance was evaluated using ( ) (R-hat) and effective sample size (ESS).
- Autocorrelation checks: Ensured independence between successive MCMC draws.
- Posterior predictive checks (PPC): Compared simulated data from posterior distributions to observed outcomes.
- Bayesian R²: Quantified the proportion of variance explained by the predictors, incorporating uncertainty.
In Bayesian analysis, every unknown parameter — such as a regression coefficient, mean, or variance — is treated as a random variable with a probability distribution that expresses uncertainty given the observed data.
Analysis and Results
Statistical Tool
R packages and libraries are used to import data, perform data wrangling and analysis.
Data source
- NHANES 2-year data (2013-2014) cross-sectional weighted data Center for Health Statistics (1999) was imported in R
Data pre-processing
Adult dataset:Three NHANES datasets (demographics, exam, questionnaire) in.XPT format are imported (Haven package) in R. Variables of interest are selected using the original weighted datasets and ID to create a single adult analytic dataframe.
Data Variables
Response Variable - Binary Type 2 / diagnosed diabetes(excluding gestational diabetes) diabetes_dx created combning - DIQ010 - Doctor told you have diabetes - DIQ050- excluded (a secondary variable describing treatment status (insulin use)). Predictor Variables - Body Mass Index, factor, 4 levels
Covariates - Gender (factor, 2 levels) - Ethnicity (factor, 5 levels) - Age (continuous 20-80years)
Exploratory Data Analysis (Adult, 20 - 80 years)
- Discrete vs Continuous Columns: 25% of columns are discrete (categorical) while 75% are continuous, indicating that the dataset primarily contains continuous measurements such as age and BMI.
- Complete Rows: 92.7% of rows have complete information for all variables, meaning most participants have fully observed data across predictors and outcomes.
Survey design: - It is a national survey based on complex sampling designs (oversampling certain groups (e.g., minorities, older adults) to ensure representation. - They use multistage sampling to represent the U.S. population, so we apply sampling weights, strata, and PSU (primary sampling units) for valid estimates. - We use survey design in regression anlaysis to avoid to avoid bias prevalence estimates (e.g., mean BMI or diabetes %), underestimation of standard errors and incorrect inference for population-level parameters. - It includes auxillary variables: SDMVPSU, SDMVSTRA, WTMEC2YR - - Diabetes grouped from (DIQ010 excluding DIQ050): diabetes_dx (numeric 0/1) - Covariates: ethnicity (5 levels), age range (20-80 years), gender (male and female), BMI as continuous - Centered covariates: age_c, bmi_c BMI categories: bmi_cat - Presented here is the mean, standard error and variance of the survey weighted data
| Step | Description |
|---|---|
| Weighting | Used the survey package to calculate weighted means and standard deviations for all variables. |
| Standardization | Standardized BMI and age variables for analysis. |
| Age Categorization | Recoded into intervals: 20–<30, 30–<40, 40–<50, 50–<60, 60–<70, and 70–80 years. |
| BMI Categorization | Recoded and categorized as: <18.5 (Underweight), 18.5–<25 (Normal), 25–<30 (Overweight), 30–<35 (Obesity I), 35–<40 (Obesity II), ≥40 (Obesity III). |
| Ethnicity Recoding | Recoded as: 1 = Mexican American, 2 = Other Hispanic, 3 = Non-Hispanic White, 4 = Non-Hispanic Black, 5 = Other/Multi. |
| Special Codes | Special codes (e.g., 3, 7) were transformed to NA. These codes are not random and could introduce bias if ignored (MAR or MNAR). |
| Missing Data | Missing values were retained and visualized to assess their pattern and informativeness. |
| Final Dataset | Created a cleaned analytic dataset (adult) using Non-Hispanic White and Male as reference groups for analysis. |
Code
##
# ---------------- Basic Exploration (adults) ----------------
# Keep adults only and build analysis variables
adult <- merged_data %>%
dplyr::filter(RIDAGEYR >= 20) %>%
dplyr::transmute(
# --- keep survey design variables so svydesign() can see them ---
SDMVPSU, SDMVSTRA, WTMEC2YR,
# --- outcome: DIQ010 (1 yes, 2 no; 3/7/9 -> NA) ---
diabetes_dx = dplyr::case_when(
DIQ010 == 1 ~ 1,
DIQ010 == 2 ~ 0,
DIQ010 %in% c(3, 7, 9) ~ NA_real_,
TRUE ~ NA_real_
),
# --- predictors (raw) ---
bmi = BMXBMI,
age = RIDAGEYR,
# sex (1=Male, 2=Female)
sex = forcats::fct_recode(factor(RIAGENDR), Male = "1", Female = "2"),
# race (5-level)
race = forcats::fct_recode(
factor(RIDRETH1),
"Mexican American" = "1",
"Other Hispanic" = "2",
"NH White" = "3",
"NH Black" = "4",
"Other/Multi" = "5"
),
# keep DIQ050 so we can safely reference it (may be absent/NA in some rows)
DIQ050 = DIQ050
) %>%
# standardize continuous predictors
dplyr::mutate(
age_c = as.numeric(scale(age)),
bmi_c = as.numeric(scale(bmi)),
bmi_cat = cut(
bmi,
breaks = c(-Inf, 18.5, 25, 30, 35, 40, Inf),
labels = c("<18.5","18.5–<25","25–<30","30–<35","35–<40","≥40"),
right = FALSE
)
) %>%
# adjust outcome: if female & DIQ050==1 ("only when pregnant"), set to 0 (not diabetes)
dplyr::mutate(
diabetes_dx = ifelse(sex == "Female" & !is.na(DIQ050) & DIQ050 == 1, 0, diabetes_dx)
)
# Make NH White the reference level for race (clearer interpretation)
adult <- adult %>%
dplyr::mutate(
race = forcats::fct_relevel(race, "NH White")
)
# --- sanity checks ---
cat("Adults n =", nrow(adult), "\n")Adults n = 5769 Code
library(dplyr)
library(skimr)
library(knitr)
library(tidyr)
library(purrr)
library(forcats)
library(kableExtra)
str(adult)'data.frame': 5769 obs. of 12 variables:
$ SDMVPSU : num 1 1 1 2 1 1 2 1 2 2 ...
$ SDMVSTRA : num 112 108 109 116 111 114 106 112 112 113 ...
$ WTMEC2YR : num 13481 24472 57193 65542 25345 ...
$ diabetes_dx: num 1 1 1 0 0 0 0 0 0 0 ...
$ bmi : num 26.7 28.6 28.9 19.7 41.7 35.7 NA 26.5 22 20.3 ...
$ age : num 69 54 72 73 56 61 42 56 65 26 ...
$ sex : Factor w/ 2 levels "Male","Female": 1 1 1 2 1 2 1 2 1 2 ...
$ race : Factor w/ 5 levels "NH White","Mexican American",..: 4 1 1 1 2 1 3 1 1 1 ...
$ DIQ050 : num 1 1 1 2 2 2 2 2 2 2 ...
$ age_c : num 1.132 0.278 1.303 1.36 0.392 ...
$ bmi_c : num -0.3359 -0.0703 -0.0283 -1.3144 1.761 ...
$ bmi_cat : Factor w/ 6 levels "<18.5","18.5–<25",..: 3 3 3 2 6 5 NA 3 2 2 ...Code
plot_str(adult)
head(adult) SDMVPSU SDMVSTRA WTMEC2YR diabetes_dx bmi age sex race DIQ050
1 1 112 13481.04 1 26.7 69 Male NH Black 1
2 1 108 24471.77 1 28.6 54 Male NH White 1
3 1 109 57193.29 1 28.9 72 Male NH White 1
4 2 116 65541.87 0 19.7 73 Female NH White 2
5 1 111 25344.99 0 41.7 56 Male Mexican American 2
6 1 114 61758.65 0 35.7 61 Female NH White 2
age_c bmi_c bmi_cat
1 1.1324183 -0.33588609 25–<30
2 0.2783598 -0.07028101 25–<30
3 1.3032300 -0.02834336 25–<30
4 1.3601672 -1.31443114 18.5–<25
5 0.3922343 1.76099614 ≥40
6 0.6769204 0.92224325 35–<40Code
plot_intro(adult, title="Figure 1 (Adult dataset). Structure of variables and missing observations.")
Code
plot_missing(adult, title="Figure 2(Adult dataset). Breakdown of missing observations.")
Study population (Adult, NHANES)
- Number of participants in Adult dataset (n = 5769) after cleaning and analysis Variables mean SE age 47.496 0.3805 mean SE diabetes_dx 0.089016 0.0048 variance SE diabetes_dx 4759.9 0.0039 Effective sample size for diabetes_dx: 48142
Population characterisitcs
- Age: Participants are fairly evenly distributed across adult age groups, with no sharp skewness.
- Sex: sample includes a higher proportion of females than males.
- BMI: Most participants have BMI values within the normal to overweight range, with fewer in the obese category.
- BMI by Diabetes Status: Individuals diagnosed with diabetes tend to have higher BMI values compared to non-diabetics.
- Diabetes Prevalence by Age Group: The proportion of diabetes increases with advancing age, highlighting age as a strong risk factor.
- Diabetes Prevalence by Race/Ethnicity: Differences are observed across racial/ethnic groups, with some showing higher prevalence rates than others.
Visualization - Below are histogram, bar graph, boxplot to display age, bmi and their association with diabetes status. Plot below shows males and females with and without diabetes (including missing data) across different racial groups. Bars are side by side for each sex, with counts displayed on top
Code
ggplot(adult, aes(x = age)) +
geom_histogram(binwidth = 5, fill = "skyblue", color = "white") +
labs(
title = "Distribution of Age >20 years",
x = "Age (years)",
y = "Count"
) +
theme_minimal()
Code
ggplot(adult, aes(factor(diabetes_dx))) +
geom_bar(fill = "steelblue") +
labs(title="Diabetes Outcome Distribution in >20 years age group", x="diabetes_dx (0=No, 1=Yes)", y="Count")
Code
ggplot(adult, aes(factor(bmi_cat))) +
geom_bar(fill = "steelblue") +
labs(title="Diabetes Outcome Distribution by BMI in >20 years age group", x="bmi_cat")
Code
ggplot(adult, aes(x = factor(diabetes_dx), y = bmi)) +
geom_boxplot(fill = "skyblue") +
labs(
title = "BMI Distribution by Diabetes Diagnosis in >20 years age group",
x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
y = "BMI"
) +
theme_minimal()
Code
# plots for adult data bmi categories and race categories
ggplot(adult, aes(x = factor(race), fill = factor(diabetes_dx))) +
geom_bar(position = "dodge") +
labs(
title = "Diabetes Diagnosis by Race in >20 years age group",
x = "Race/Ethnicity",
y = "Count",
fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
Code
ggplot(adult, aes(x = factor(bmi_cat), fill = factor(diabetes_dx))) +
geom_bar(position = "dodge") +
labs(
title = "Diabetes Diagnosis by BMI in >20 years age group",
x = "BMI",
y = "Count",
fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
Code
# Example: create your dataset
adult1 <- data.frame(
race = rep(c("NH White","Mexican American","Other Hispanic","NH Black","Other/Multi"), each = 6),
sex = rep(c("Male","Male","Male","Female","Female","Female"), times = 5),
diabetes_dx = rep(c(0,1,NA,0,1,NA), times = 5),
count = c(
1019,119,38,1164,96,36,
304,60,14,329,49,11,
183,26,10,255,25,9,
461,100,19,515,65,17,
351,46,8,393,32,15
)
)
# Clean NA for plotting or convert to "Missing"
adult1$diabetes_dx <- as.character(adult1$diabetes_dx)
adult1$diabetes_dx[is.na(adult1$diabetes_dx)] <- "Missing"
# Plot grouped bar chart
ggplot(adult1, aes(x = diabetes_dx, y = count, fill = sex)) +
geom_bar(stat = "identity", position = "dodge") +
facet_wrap(~race) +
labs(title = "Diabetes Diagnosis by Sex and Race",
x = "Diabetes Diagnosis",
y = "Count") +
theme_minimal() +
scale_fill_manual(values = c("skyblue", "orange"))
Abnormalities detected in Adult dataset
Missingness
- Only 1.3% of individual data points are missing across the dataset, reflecting minimal missingness.
- No column is entirely missing (0%), indicating all variables have at least some observed data.
- Overall missingness: ~4% → low, but non-trivial given the small number of variables involved.
- Missingness is not completely at random (MNAR or MAR) - If the probability of missingness depends on other observed variables (e.g., older adults missing BMI due to illness), imputation helps reduce bias. It is possible and should consider MICE and test with logistic regression of missingness indicators
- Missingness affects outcome or key covariates - Even small missingness in important variables can bias posterior estimates. Since BMI and diabetes are central we should perform MICE
- Sufficient auxiliary variables available - MICE works best when you have other correlated variables to inform imputation (e.g., age, sex, race, WTMEC2YR).
- Bayesian model assumes complete data - Standard Bayesian logistic models (e.g., brms, rstanarm) cannot directly handle NAs — you must impute or model missingness.
Statistical Modeling Approach
Data Preparation and Survey Design Specification
-Survey design variables — primary sampling unit (SDMVPSU), strata (SDMVSTRA), and examination weights (WTMEC2YR) — were retained to account for the complex, multistage sampling of NHANES. - A survey design object ensures a population-representative estimates and valid variance estimation. - Frequentist Survey-Weighted Logistic Regression (Complete-Case Analysis) - fitted using only complete cases to assess the associations between diabetes status (binary outcome) and predictors such as BMI, age, sex, and race/ethnicity. - Survey weights were applied to correct for unequal probabilities of selection and nonresponse, ensuring generalizability to the U.S. adult population.
Handling Missing Data: Multivariate Imputation by Chained Equations (MICE)
- Performed to address missing values in BMI and other covariates.
- Variable with missing data imputed conditionally on all others through iterative regression models.
- Multiple (m = 5–10) imputed datasets generated, were analyzed separately, and combined using Rubin’s rules to obtain pooled parameter estimates and standard errors.
Bayesian Logistic Regression (Post-Imputation Analysis)
- Bayesian logistic regression model applied to the imputed datasets, incorporated prior distributions for regression coefficients and allowed direct estimation of posterior distributions, credible intervals, and posterior predictive checks.
- Bayesian inference provided a probabilistic interpretation of parameter uncertainty, complementing the frequentist findings.
Model Validation and Interpretation
- Diagnostic checks performed below evaluate model convergence, goodness-of-fit, and predictive accuracy.
Model Comparison
- The results from both frameworks (frequentist and Bayesian) were compared to ensure robustness of conclusions regarding predictors of diabetes.
| term | OR | LCL | UCL | p.value |
|---|---|---|---|---|
| age_c | 3.0292807 | 2.6967690 | 3.4027912 | 0.0000000 |
| bmi_c | 1.8853571 | 1.6526296 | 2.1508579 | 0.0000039 |
| sexFemale | 0.5281132 | 0.4104905 | 0.6794397 | 0.0003857 |
| raceMexican American | 2.0358434 | 1.4850041 | 2.7910081 | 0.0008262 |
| raceOther Hispanic | 1.5915182 | 1.1664529 | 2.1714810 | 0.0087119 |
| raceNH Black | 1.6689718 | 1.1605895 | 2.4000450 | 0.0116773 |
| raceOther/Multi | 2.3270527 | 1.5451752 | 3.5045697 | 0.0014331 |
Summary
Multiple Logistic Regression model (Survey weighted) - Identified several significant predictors of diabetes diagnosis after adjusting for demographic and anthropometric factors. - Age (OR = 2.90, 95% CI: 2.60–3.24, p < 0.001): Older adults had nearly three times higher odds of diabetes compared with younger participants, indicating a strong positive association between age and diabetes risk. - BMI (OR = 1.73, 95% CI: 1.58–1.89, p < 0.001): Higher body mass index was significantly associated with increased odds of diabetes, confirming obesity as a key risk factor. - Sex (Female vs. Male: OR = 0.54, 95% CI: 0.45–0.65, p < 0.001): Females had significantly lower odds of diabetes compared to males. - Race/Ethnicity: Mexican American (OR = 2.43, 95% CI: 1.86–3.18, p < 0.001) Other Hispanic (OR = 1.75, 95% CI: 1.24–2.47, p = 0.001) Non-Hispanic Black (OR = 1.98, 95% CI: 1.56–2.50, p < 0.001) Other/Multi-racial (OR = 2.11, 95% CI: 1.56–2.85, p < 0.001) All minority racial/ethnic groups had significantly higher odds of diabetes compared with the reference group (Non-Hispanic Whites).
Multivariate Imputation by Chained Equations (Pooled Logistic Regression)
- We conducted MICE to manage missiging data as an alternative to the Bayesian Approach Buuren and Groothuis-Oudshoorn (2011)
- Flatness of the density, heavy tails, non-zero peakedness, skewness and multimodality do not hamper the good performance of multiple imputation for the mean structure in samples n > 400 even for high percentages (75%) of missing data in one variable Van Buuren and Van Buuren (2012).
- Multiple Imputation (MI) can be performed using mice package in R
- Iterative mice imputes missing values of one variable at a time, using regression models based on the other variables in the dataset.
- In the chain process, each imputed variable become a predictor for the subsequent imputation, and the entire process is repeated multiple times to create several complete datasets, each reflecting different possibilities for the missing data.
Code
# ----- Multiple Imputation (predictors only)
mi_dat <- adult %>%
dplyr::select(diabetes_dx, age, bmi, sex, race, WTMEC2YR, SDMVPSU, SDMVSTRA)
meth <- mice::make.method(mi_dat)
pred <- mice::make.predictorMatrix(mi_dat)
# Do not impute outcome
meth["diabetes_dx"] <- ""
pred["diabetes_dx", ] <- 0
pred[,"diabetes_dx"] <- 1
# Imputation methods
meth["age"] <- "norm"
meth["bmi"] <- "pmm"
meth["sex"] <- "polyreg"
meth["race"] <- "polyreg"
# Survey design vars as auxiliaries only
meth[c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- ""
pred[, c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- 1
glimpse(mi_dat)Rows: 5,769
Columns: 8
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, NA, 26.5, 22.0, 20.3, …
$ sex <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…Code
imp <- mice::mice(mi_dat, m = 5, method = meth, predictorMatrix = pred, seed = 123)
iter imp variable
1 1 bmi
1 2 bmi
1 3 bmi
1 4 bmi
1 5 bmi
2 1 bmi
2 2 bmi
2 3 bmi
2 4 bmi
2 5 bmi
3 1 bmi
3 2 bmi
3 3 bmi
3 4 bmi
3 5 bmi
4 1 bmi
4 2 bmi
4 3 bmi
4 4 bmi
4 5 bmi
5 1 bmi
5 2 bmi
5 3 bmi
5 4 bmi
5 5 bmiResults: MICE (pooled imputed dataset): - In the final analytic dataset consisting of 5,769 participants with 8 variables, with missing values were addressed using Multivariate Imputation by Chained Equations (MICE). - Five imputations were performed across five iterations each, with BMI imputed conditionally based on other predictors (age, sex, race, and diabetes status). - The iterative process showed stable convergence, indicating reliable estimation of missing BMI values for subsequent survey-weighted and Bayesian modeling analyses.
Code
fit_mi <- with(imp, {
age_c <- as.numeric(scale(age))
bmi_c <- as.numeric(scale(bmi))
glm(diabetes_dx ~ age_c + bmi_c + sex + race, family = binomial())
})
pool_mi <- pool(fit_mi)
summary(pool_mi) term estimate std.error statistic df p.value
1 (Intercept) -2.6895645 0.09941301 -27.054453 5566.204 1.486581e-151
2 age_c 1.0660265 0.05594733 19.054108 5520.446 1.911564e-78
3 bmi_c 0.5468538 0.04473386 12.224604 5148.557 6.751227e-34
4 sexFemale -0.6178297 0.09379129 -6.587282 5551.660 4.892566e-11
5 raceMexican American 0.8877355 0.13750463 6.456041 5472.583 1.167455e-10
6 raceOther Hispanic 0.5606621 0.17485537 3.206433 5573.987 1.351505e-03
7 raceNH Black 0.6809629 0.11981185 5.683602 5576.734 1.385727e-08
8 raceOther/Multi 0.7476406 0.15300663 4.886328 4749.963 1.061140e-06Code
## table
mi_or <- summary(pool_mi, conf.int = TRUE, exponentiate = TRUE) %>%
dplyr::rename(
term = term, OR = estimate, LCL = `2.5 %`, UCL = `97.5 %`, p.value = p.value
) %>%
dplyr::filter(term != "(Intercept)")
knitr::kable(mi_or, caption = "MI pooled odds ratios (per 1 SD)")| term | OR | std.error | statistic | df | p.value | LCL | UCL | conf.low | conf.high | |
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | age_c | 2.9038183 | 0.0559473 | 19.054108 | 5520.446 | 0.0000000 | 2.6021752 | 3.2404277 | 2.6021752 | 3.2404277 |
| 3 | bmi_c | 1.7278084 | 0.0447339 | 12.224604 | 5148.557 | 0.0000000 | 1.5827382 | 1.8861754 | 1.5827382 | 1.8861754 |
| 4 | sexFemale | 0.5391132 | 0.0937913 | -6.587282 | 5551.660 | 0.0000000 | 0.4485669 | 0.6479368 | 0.4485669 | 0.6479368 |
| 5 | raceMexican American | 2.4296216 | 0.1375046 | 6.456041 | 5472.583 | 0.0000000 | 1.8555327 | 3.1813298 | 1.8555327 | 3.1813298 |
| 6 | raceOther Hispanic | 1.7518320 | 0.1748554 | 3.206433 | 5573.987 | 0.0013515 | 1.2434346 | 2.4680953 | 1.2434346 | 2.4680953 |
| 7 | raceNH Black | 1.9757793 | 0.1198118 | 5.683602 | 5576.734 | 0.0000000 | 1.5621842 | 2.4988753 | 1.5621842 | 2.4988753 |
| 8 | raceOther/Multi | 2.1120110 | 0.1530066 | 4.886328 | 4749.963 | 0.0000011 | 1.5646727 | 2.8508138 | 1.5646727 | 2.8508138 |
- Age remained the strongest predictor of diabetes (OR = 2.90, 95% CI: 2.60–3.24, p < 0.001).
- BMI continued to show a significant positive association (OR = 1.73, 95% CI: 1.58–1.89, p < 0.001).
- Female sex was associated with lower odds of diabetes compared to males (OR = 0.54, 95% CI: 0.45–0.65, p < 0.001).
- Compared to Non-Hispanic Whites, higher odds of diabetes were observed among: Mexican Americans (OR = 2.43, 95% CI: 1.86–3.18) Other Hispanics (OR = 1.75, 95% CI: 1.24–2.47) Non-Hispanic Blacks (OR = 1.98, 95% CI: 1.56–2.50) Other/Multi-racial groups (OR = 2.11, 95% CI: 1.56–2.85)
- All associations were statistically significant (p < 0.01).
Code
library(gt)
# Bayesian Logistic Regression (formula weights)
adult_imp1 <- complete(imp, 1) %>%
dplyr::mutate(
age_c = as.numeric(scale(age)),
bmi_c = as.numeric(scale(bmi)),
wt_norm = WTMEC2YR / mean(WTMEC2YR, na.rm = TRUE),
# ensure factor refs match survey/mice:
race = forcats::fct_relevel(race, "NH White"),
sex = forcats::fct_relevel(sex, "Male")
) %>%
dplyr::filter(!is.na(diabetes_dx), !is.na(age_c), !is.na(bmi_c),
!is.na(sex), !is.na(race)) %>%
droplevels()
stopifnot(all(is.finite(adult_imp1$wt_norm)))
glimpse(adult_imp1)Rows: 5,592
Columns: 11
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, 23.6, 26.5, 22.0, 20.3…
$ sex <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…
$ age_c <dbl> 1.13241831, 0.27835981, 1.30323001, 1.36016725, 0.39223428…
$ bmi_c <dbl> -0.33319172, -0.06755778, -0.02561558, -1.31184309, 1.7639…
$ wt_norm <dbl> 0.3393916, 0.6160884, 1.4398681, 1.6500477, 0.6380722, 1.5…Code
library(tableone)
vars <- c("age", "bmi", "age_c", "bmi_c", "wt_norm", "sex", "race", "diabetes_dx")
table1 <- CreateTableOne(vars = vars, data = adult_imp1, factorVars = c("sex", "race", "diabetes_dx"))
print(table1, showAllLevels = TRUE)
level Overall
n 5592
age (mean (SD)) 48.84 (17.57)
bmi (mean (SD)) 29.00 (7.11)
age_c (mean (SD)) -0.02 (1.00)
bmi_c (mean (SD)) -0.01 (0.99)
wt_norm (mean (SD)) 1.00 (0.79)
sex (%) Male 2669 (47.7)
Female 2923 (52.3)
race (%) NH White 2398 (42.9)
Mexican American 742 (13.3)
Other Hispanic 489 ( 8.7)
NH Black 1141 (20.4)
Other/Multi 822 (14.7)
diabetes_dx (%) 0 4974 (88.9)
1 618 (11.1) Code
## correlation matrix
library(ggplot2)
library(reshape2)
correlation_matrix <- cor(adult_imp1[, c("diabetes_dx", "age", "bmi")], use = "complete.obs", method = "pearson")
correlation_melted <- melt(correlation_matrix)
ggplot(correlation_melted, aes(Var1, Var2, fill = value)) +
geom_tile(color = "white") +
scale_fill_gradient2(low = "blue", high = "red", mid = "white", midpoint = 0,
limit = c(-1, 1), space = "Lab", name = "Correlation") +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
labs(title = "Correlation Heatmap", x = "Features", y = "Features")
Visualization of imputed dataset
Pairwise correlations heatmap: show the strength and direction of correlations (Pearson correlation) which measures linear association between diabetes_dx, age, and bmi
Code
# Class distribution
ggplot(adult_imp1, aes(x = factor(diabetes_dx))) +
geom_bar(fill = "steelblue") +
labs(
title = "Diabetes Diagnosis Distribution",
x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
y = "Count"
) +
theme_minimal()
Code
prop.table(table(adult_imp1$diabetes_dx))
0 1
0.889485 0.110515 Code
# Visualization of Diabetes vs BMI (adult_data1)
library(ggplot2)
# Create the plot
ggplot(adult_imp1, aes(x = factor(diabetes_dx), y = bmi, fill = factor(diabetes_dx))) +
geom_boxplot(alpha = 0.7) +
scale_x_discrete(labels = c("0" = "No Diabetes", "1" = "Diabetes")) +
labs(
x = "Diabetes Diagnosis",
y = "BMI",
title = "BMI Distribution by Diabetes Status"
) +
theme_minimal() +
theme(legend.position = "none")
Code
# logistic regression curve
ggplot(adult_imp1, aes(x = bmi, y = diabetes_dx)) +
geom_point(aes(y = diabetes_dx), alpha = 0.2, position = position_jitter(height = 0.02)) +
geom_smooth(method = "glm", method.args = list(family = "binomial"), se = TRUE, color = "blue") +
labs(
x = "BMI",
y = "Probability of Diabetes",
title = "Predicted Probability of Diabetes vs BMI"
) +
theme_minimal()
Code
# Save your dataset as CSV
write.csv(adult_imp1, "adult_imp1.csv", row.names = FALSE)After MICE, below are the number of rows and column of the three datasets created
- Rows: 10175 and Columns: 10 (survey-weighted, merged data)
- Rows: 5,769 and Columns: 12 (filtered data, adult)
- Rows: 5,592 and Columns: 11 (imputed data, adult_imp1)
Bayesian Logistic Regression analysis on imputed dataset (adult_imp1)
- Model Overview
- A Bayesian logistic regression model was fitted on the first imputed dataset (adult_imp1) to assess predictors of diabetes diagnosis.
- Survey weights-Normalized MEC exam weights (wt_norm) with mean 1.00 (SD 0.79)
- No missing values remain in selected predictors or outcome.
- Continuous variables are standardized, which facilitates prior specification.
- Categorical variables are correctly re-leveled for reference categories.
- Weights are available for inclusion in the likelihood to account for survey design.
- Prior Specification
- to stabilize estimation in the presence of correlated predictors or outliers while retaining interpretability of model parameters.
- Intercept prior: student_t(3, 0, 10) — allowing heavy tails for flexibility in the intercept estimate. R. V. D. Schoot et al. (2013)
- Regression coefficients prior: normal(0, 2.5) — providing weakly informative regularization provide gentle regularization, constraining extreme values without overpowering the data R. van de Schoot et al. (2021)
- Model Estimation
- The model estimates using four Markov Chain Monte Carlo (MCMC) chains, each with 2000 iterations (50% warm-up), and an adaptive delta of 0.95 ensure good chain convergence and reduce divergent transitions.
- Posterior summaries represent the central tendency and uncertainty around the model parameters through credible intervals (CrI).
Code
library(gt)
priors <- c(
set_prior("normal(0, 2.5)", class = "b"),
set_prior("student_t(3, 0, 10)", class = "Intercept")
)
bayes_fit <- brm(
formula = diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race,
data = adult_imp1,
family = bernoulli(link = "logit"),
prior = priors,
chains = 4, iter = 2000, seed = 123,
control = list(adapt_delta = 0.95),
refresh = 0 # quiet Stan output
)Running /opt/R/4.4.2/lib/R/bin/R CMD SHLIB foo.c
using C compiler: ‘gcc (GCC) 11.5.0 20240719 (Red Hat 11.5.0-2)’
gcc -I"/opt/R/4.4.2/lib/R/include" -DNDEBUG -I"/opt/R/4.4.2/lib/R/library/Rcpp/include/" -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/" -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/unsupported" -I"/opt/R/4.4.2/lib/R/library/BH/include" -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/src/" -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/" -I"/opt/R/4.4.2/lib/R/library/RcppParallel/include/" -I"/opt/R/4.4.2/lib/R/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/usr/local/include -fpic -g -O2 -c foo.c -o foo.o
In file included from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Core:19,
from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Dense:1,
from /opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22,
from <command-line>:
/opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: cmath: No such file or directory
679 | #include <cmath>
| ^~~~~~~
compilation terminated.
make: *** [/opt/R/4.4.2/lib/R/etc/Makeconf:195: foo.o] Error 1Code
prior_summary(bayes_fit) prior class coef group resp dpar nlpar lb ub
normal(0, 2.5) b
normal(0, 2.5) b age_c
normal(0, 2.5) b bmi_c
normal(0, 2.5) b raceMexicanAmerican
normal(0, 2.5) b raceNHBlack
normal(0, 2.5) b raceOtherDMulti
normal(0, 2.5) b raceOtherHispanic
normal(0, 2.5) b sexFemale
student_t(3, 0, 10) Intercept
tag source
user
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
userCode
summary(bayes_fit) # Bayesian model summary Family: bernoulli
Links: mu = logit
Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race
Data: adult_imp1 (Number of observations: 5592)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -2.66 0.09 -2.84 -2.50 1.00 4187 3510
age_c 1.09 0.06 0.97 1.22 1.00 3012 3098
bmi_c 0.63 0.05 0.53 0.72 1.00 3472 3315
sexFemale -0.66 0.10 -0.86 -0.46 1.00 4003 3052
raceMexicanAmerican 0.69 0.18 0.35 1.04 1.00 3526 2843
raceOtherHispanic 0.43 0.24 -0.07 0.89 1.00 4058 3114
raceNHBlack 0.54 0.15 0.24 0.82 1.00 3597 3177
raceOtherDMulti 0.82 0.19 0.45 1.19 1.00 3763 3257
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Results and Visualization
- Age and BMI both were positively associated with diabetes risk.
- Higher standardized values corresponded to higher posterior odds of diabetes. Sex: Females showed lower odds of diabetes compared to males.
- Race/Ethnicity: Certain racial/ethnic groups demonstrated reduced odds of diabetes compared to the Non-Hispanic White reference group.
Code
library(ggplot2)
# adult_imp1 plot
# Convert to long format
adult_long <- adult_imp1 %>%
select(bmi_c, age_c) %>%
pivot_longer(cols = everything(), names_to = "Coefficient", values_to = "Value")
# Plot
ggplot(adult_long, aes(x = Value, fill = Coefficient)) +
geom_density(alpha = 0.5) +
theme_minimal() +
labs(title = "Distributions for Coefficients from adult_imp1 data",
x = "Coefficient Value", y = "Density") +
scale_fill_manual(values = c("bmi_c" = "skyblue", "age_c" = "orange"))
Code
## prior draws
prior_draws <- tibble(
term = rep(c("Age (per 1 SD)", "BMI (per 1 SD)"), each = 4000),
value = c(rnorm(4000, 0, 2.5), rnorm(4000, 0, 2.5))
)
## Plot (prior) (age and bmi)
ggplot(prior_draws, aes(x = value, fill = term)) +
geom_density(alpha = 0.5) +
theme_minimal() +
labs(title = "Prior Distributions for Coefficients",
x = "Coefficient Value", y = "Density") +
scale_fill_manual(values = c("skyblue", "orange"))
Visualization of Priors and Data Distributions
- Data-Derived Coefficient Distributions
- The density plots of standardized BMI and Age from the imputed dataset (adult_imp1) show approximately normal distributions centered near zero, consistent with z-score standardization confirming that both predictors were properly centered and scaled prior to Bayesian modeling, ensuring comparability and numerical stability during estimation.
- Prior Distributions
- Priors for regression coefficients were drawn from a Normal(0, 2.5) distribution, representing weakly informative assumptions centered at zero with moderate spread. The prior density plots for Age (per 1 SD) and BMI (per 1 SD) demonstrate symmetric bell-shaped distributions, indicating no strong bias toward positive or negative effects before observing data.
Predictive checking and validation (Bayesian model)
- Posterior Summaries (mean, median, 95% credible intervals)
- Convergence diagnostics (R-hat, effective sample size)
- plots to visualizes posterior distributions with high uncertainty, narrow distributions indicating precise estimates.
- Posterior Odds Ratios provides interpretation of the model coefficients on a multiplicative scale with reference categories: NH White (race), Male (sex).
- Posterior Predictive Checks (PPC) assesses how the model reproduces observed data and validate model fit.
- Visualizations of generated simulated datasets compared with the observed data show density overlays for both mean and SD. There was no large discrepancies indicating potential misfit; there was good alignment suggesting reliable predictions.
- MCMC Convergence endures reliable posterior estimates.
- MCMC Trace plots show chains for each parameter over iterations.
- Well-mixed chains without trends indicate convergence and stable posterior estimates.
- Model Fit -provided details to quantify predictive performance.
- The proportion of variance explained by the model: R² = 0.13 (13%) shows predictors are relevant but other factors (e.g., genetics, lifestyle, environment) also contribute to outcome variability.
- Correlation and Parameter Relationships (Optional)
- Pairwise plots (mcmc_pairs, posterior) – explore correlations between parameters.
- Histograms or density plots mcmc_hist() or mcmc_areas() of specific parameters detects no collinearity or dependencies among predictors
Code
library(brms)
summary(bayes_fit) Family: bernoulli
Links: mu = logit
Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race
Data: adult_imp1 (Number of observations: 5592)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -2.66 0.09 -2.84 -2.50 1.00 4187 3510
age_c 1.09 0.06 0.97 1.22 1.00 3012 3098
bmi_c 0.63 0.05 0.53 0.72 1.00 3472 3315
sexFemale -0.66 0.10 -0.86 -0.46 1.00 4003 3052
raceMexicanAmerican 0.69 0.18 0.35 1.04 1.00 3526 2843
raceOtherHispanic 0.43 0.24 -0.07 0.89 1.00 4058 3114
raceNHBlack 0.54 0.15 0.24 0.82 1.00 3597 3177
raceOtherDMulti 0.82 0.19 0.45 1.19 1.00 3763 3257
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Code
plot(bayes_fit) # Posterior distributions
Code
mcmc_trace(bayes_fit) # Convergence (optional)
Code
bayes_R2(bayes_fit) # Model fit Estimate Est.Error Q2.5 Q97.5
R2 0.1313342 0.01265055 0.1064607 0.156078Code
# Posterior ORs (drop intercept, clean labels)
bayes_or <- posterior_summary(bayes_fit, pars = "^b_") %>%
as.data.frame() %>%
tibble::rownames_to_column("raw") %>%
dplyr::mutate(
term = gsub("^b_", "", raw),
term = gsub("race", "race:", term),
term = gsub("sex", "sex:", term),
term = gsub("OtherDMulti", "Other/Multi", term),
term = gsub("OtherHispanic", "Other Hispanic", term),
OR = exp(Estimate),
LCL = exp(Q2.5),
UCL = exp(Q97.5)
) %>%
dplyr::select(term, OR, LCL, UCL) %>%
dplyr::filter(term != "Intercept")
knitr::kable(
bayes_or %>%
dplyr::mutate(dplyr::across(c(OR,LCL,UCL), ~round(.x, 2))),
digits = 2,
caption = "Bayesian posterior odds ratios (95% CrI) — reference: NH White (race), Male (sex)"
)| term | OR | LCL | UCL |
|---|---|---|---|
| age_c | 2.99 | 2.64 | 3.37 |
| bmi_c | 1.87 | 1.71 | 2.05 |
| sex:Female | 0.52 | 0.42 | 0.63 |
| race:MexicanAmerican | 2.00 | 1.41 | 2.84 |
| race:Other Hispanic | 1.54 | 0.93 | 2.43 |
| race:NHBlack | 1.71 | 1.28 | 2.27 |
| race:Other/Multi | 2.27 | 1.56 | 3.28 |
Compact table compare odds ratios (ORs) and 95% confidence/credible intervals (CIs) across three models for BMI and Age - Survey-weighted maximum likelihood estimation (MLE) - Multiple imputation pooled estimates (mice) - Bayesian Logistic Regression
Code
# Results
#Build compact results table (BMI & Age only)
library(dplyr);
library(tidyr);
library(knitr);
library(stringr)
# pretty "OR (LCL–UCL)" string
fmt_or <- function(or, lcl, ucl, digits = 2) {
paste0(
formatC(or, format = "f", digits = digits), " (",
formatC(lcl, format = "f", digits = digits), "–",
formatC(ucl, format = "f", digits = digits), ")"
)
}
# guardrails: require these to exist from Modeling
stopifnot(exists("svy_or"), exists("mi_or"), exists("bayes_or"))
for (nm in c("svy_or","mi_or","bayes_or")) {
if (!all(c("term","OR","LCL","UCL") %in% names(get(nm)))) {
stop(nm, " must have columns: term, OR, LCL, UCL")
}
}
svy_tbl <- svy_or %>% mutate(Model = "Survey-weighted MLE")
mi_tbl <- mi_or %>% mutate(Model = "mice pooled")
bayes_tbl <- bayes_or %>% mutate(Model = "Bayesian")
all_tbl <- bind_rows(svy_tbl, mi_tbl, bayes_tbl) %>%
mutate(term = case_when(
str_detect(term, "bmi_c|\\bBMI\\b") ~ "BMI (per 1 SD)",
str_detect(term, "age_c|\\bAge\\b") ~ "Age (per 1 SD)",
TRUE ~ term
)) %>%
filter(term %in% c("BMI (per 1 SD)", "Age (per 1 SD)")) %>%
mutate(OR_CI = fmt_or(OR, LCL, UCL, digits = 2)) %>%
select(Model, term, OR_CI) %>%
arrange(
factor(Model, levels = c("Survey-weighted MLE","mice pooled","Bayesian")),
factor(term, levels = c("BMI (per 1 SD)","Age (per 1 SD)"))
)
res_wide <- all_tbl %>%
pivot_wider(names_from = term, values_from = OR_CI) %>%
rename(
`BMI (per 1 SD) OR (95% CI)` = `BMI (per 1 SD)`,
`Age (per 1 SD) OR (95% CI)` = `Age (per 1 SD)`
)
kable(
res_wide,
align = c("l","c","c"),
caption = "Odds ratios (per 1 SD) with 95% CIs across models"
)| Model | BMI (per 1 SD) OR (95% CI) | Age (per 1 SD) OR (95% CI) |
|---|---|---|
| Survey-weighted MLE | 1.89 (1.65–2.15) | 3.03 (2.70–3.40) |
| mice pooled | 1.73 (1.58–1.89) | 2.90 (2.60–3.24) |
| Bayesian | 1.87 (1.71–2.05) | 2.99 (2.64–3.37) |
Code
# Posterior predictive draws
#Posterior predictive checks (binary outcome)
pp_samples <- posterior_predict(bayes_fit, ndraws = 500) # 500 draws
# Check dimensions
dim(pp_samples) # rows = draws, cols = observations[1] 500 5592Code
# Plot overlay of observed vs predicted counts (duplicate image)
ppc_dens_overlay(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:50, ]) +
labs(title = "Posterior Predictive Check: Density Overlay") +
theme_minimal()
Code
ppc_bars(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:50, ])
Code
#PP check for proportions (useful for binary) mean and sd comparison to check if the simulated means match the observed mean
## mean
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "mean") +
labs(title = "Posterior Predictive Check: Mean of Replicates") +
theme_minimal()
Code
## sd
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "sd") +
labs(title = "PPC: Standard Deviation of Replicates") +
theme_minimal()
Comparative Visualizations
- Predicted vs observed - to check how well the model’s predictions align with reality where mean(y_rep) = average predicted probability of diabetes for each individual, across all posterior draws of the parameters. y = the actual observed diabetes status (0 = non-diabetic, 1 = diabetic).
- mcmc dens plots - compare observed and posterior parameter values (estimates) for bmi_c, age_c, sex_female, and by race categories
- Fitted (Predicted) vs observed for bmi using point and error bars
- Fitted (Predicted) vs observed for bmi using line plot
Comparative Visualizations for Model Assessment
Visualization to evaluate the Bayesian model predicts of diabetes-related outcomes: shows comparative plots of observed and posterior-predicted values
- Posterior Parameter Distributions assess uncertainty and effect sizes of predictors, we extracted posterior draws using as_draws_df(bayes_fit).
- Plotted density overlays (mcmc_areas) for key predictors: age, BMI, sex, and race categories showing distribution of posterior estimates to visualize uncertainty and parameter magnitude.
- Predicted vs Observed Values to evaluate model fit by comparing predicted outcomes to actual data using fitted (bayes_fit, summary = TRUE).
- Compared with the observed values for continuous predictors (e.g., BMI and age).
- Visualization in Scatter plots with point estimates and 95% credible intervals as error bars shows Line of perfect agreement (slope = 1, dashed red line) for reference.
- Scatter and error bar plots indicate predicted BMI values align with observed BMI across individuals.
- Good alignment along the 45° line suggests reliable predictions; deviations highlight areas where the model may under- or over-predict.
- Overall, these visualizations complement posterior summaries and predictive checks, supporting model validation and interpretation.
Predicted vs Observed BMI
To evaluate model fit at the individual level, we compared observed BMI values to posterior-predicted BMI estimates from the Bayesian model: - A comaprative results from observed BMI (bmi) and predicted posterior estimates (predicted_bmi) with 95% credible intervals (lower_ci, upper_ci) from the posterior draws in a Line plot of BMI over observations: - Observed BMI: solid line. - Predicted BMI: solid line of posterior means. - Shaded ribbon: 95% credible interval around predicted values to visualize uncertainty. - Close alignment of predicted lines with observed BMI indicates good model fit. - Wider ribbons highlight greater posterior uncertainty for individual predictions. - Summary statistics for bmi and standardized bmi_c help contextualize the observed range and variability in the sample.
Code
# Combine observed and predicted into one data frame
plot_data <- adult_imp1 %>%
mutate(
predicted_bmi = predicted[, "Estimate"],
lower_ci = predicted[, "Q2.5"],
upper_ci = predicted[, "Q97.5"],
obs_index = 1:nrow(adult_imp1) # index for x-axis
)
# Line plot
ggplot(plot_data, aes(x = obs_index)) +
geom_line(aes(y = bmi, color = "Observed")) + # observed BMI
geom_line(aes(y = predicted_bmi, color = "Predicted")) + # predicted BMI
geom_ribbon(aes(ymin = lower_ci, ymax = upper_ci), alpha = 0.2) + # uncertainty
labs(x = "Observation", y = "BMI", color = "Legend") +
theme_minimal()
Code
summary(adult_imp1$bmi) Min. 1st Qu. Median Mean 3rd Qu. Max.
14.1 24.1 27.7 29.0 32.4 82.9 Code
summary(plot_data$bmi_c) Min. 1st Qu. Median Mean 3rd Qu. Max.
-2.09476 -0.69669 -0.19338 -0.01133 0.46371 7.52398 Summary of observed data and centered BMI - Centering doesn’t change the distribution shape, only shifts it so the mean is zero. - Centering is useful in regression/Bayesian models to improve numerical stability and interpretability of intercepts - Plots showing predicted values of bmi and age (prior vs predicted) and the proportion of diabetes=1 for each draw
Visualization on Prior vs Posterior Distributions
- To assess how the Bayesian model updates beliefs from prior information to posterior estimates, we compared prior vs posterior coefficient distributions for key predictors: BMI and age.
- Prior Draws
- Simulated from a standard normal distribution (mean = 0, SD = 1) for both BMI and age coefficients. Represent initial beliefs about coefficient values before seeing the data.
- Posterior Draws
- Extracted from the fitted model (bayes_fit) for b_bmi_c and b_age_c.
- Pivoted to long format and labeled as “Posterior”.
- Visualization Combined prior and posterior draws
- Plotted density overlays with facets for BMI and age.
- Posterior distributions are narrower and often shifted from prior, reflecting information gained from the data.
- Differences between prior and posterior highlight the model’s learning about effect sizes.
- Posterior Predictive Proportions of Diabetes
- Computed the proportion of diabetes cases (diabetes = 1) for each posterior draw (pp_samples).
Interpretaion: - Prior vs posterior plots demonstrate that the Bayesian model updates prior beliefs in a data-informed way. - Posterior predictive proportions closely match observed prevalence, supporting model reliability for inference and prediction.
Code
library(ggplot2)
ggplot(combined_draws, aes(x = estimate, fill = type)) +
geom_density(alpha = 0.4) +
facet_wrap(~ term, scales = "free", ncol = 2) +
theme_minimal(base_size = 13) +
labs(
title = "Prior vs Posterior Distributions",
x = "Coefficient estimate",
y = "Density",
fill = ""
)
Code
# Compute proportion of diabetes=1 for each draw
pp_proportion <- rowMeans(pp_samples) # proportion of 1's in each posterior draw
# Summary of posterior proportions
summary(pp_proportion) Min. 1st Qu. Median Mean 3rd Qu. Max.
0.08959 0.10443 0.10882 0.10892 0.11360 0.12697 Code
# Optional: visualize the posterior probability distribution
pp_proportion_df <- tibble(proportion = pp_proportion)
ggplot(pp_proportion_df, aes(x = proportion)) +
geom_histogram(binwidth = 0.01, fill = "skyblue", color = "black") +
labs(
title = "Posterior Distribution of Proportion of Diabetes = 1",
x = "Proportion of Diabetes = 1",
y = "Frequency"
) +
theme_minimal()
Code
svy_mean <- svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)| Method | diabetes_mean | SE |
|---|---|---|
| Survey-weighted mean (NHANES) | 0.0890 | 0.0048 |
| Imputed dataset mean | 0.1105 | NA |
| Posterior predictive mean | 0.1089 | NA |
Comaprison of Prior and Predicted draws for both Age and BMI
- Plot shows the posterior distributions are much more concentrated around ~0.5 (example) than the priors indicating the data provided strong evidence about the effect size of these covariates.
- The priors were diffuse, showing initial uncertainty
- The posteriors are precise, showing learning from the data.
Propotion of diabetes in the posterior draws
States from the predicted probability of diabetes = 1 in the population (Bayesian model). -Min = 0.085 → In some posterior draws, only ~8.5% of the population is predicted to have diabetes. -1st Quartile = 0.105 → 25% of posterior draws predict diabetes prevalence below 10.5%. -Median = 0.109 → Half of the simulated draws predict a prevalence below ~10.9%, half above. -Mean = 0.109 → The average predicted prevalence is ~10.9%, very close to the median → roughly symmetric distribution. -3rd Quartile = 0.113 → 75% of draws predict prevalence below ~11.3%. -Max = 0.128 → The highest predicted prevalence across all draws is ~12.8%.
Bayesian model predicts that about 10–11% of this population has diabetes, with a relatively narrow range across posterior draws, reflects uncertainty in the estimate
While most predictions cluster around 10–11%, the model allows for values as low as 8.5% and as high as 12.8%.
On comparing this with the raw imputed data proportion show that the the model predictions align with the observed/imputed data.
Clinical relevance of predicted proportion - it accounts for uncertainty in the model and imputed data. - Policy makers or clinicians could plan interventions accordingly anticipating ~1 in 10 adults in this population might have diabetes.
Code
library(tidyverse)
# Posterior predicted proportion vector
# pp_proportion <- rowMeans(pp_samples) # if not already done
known_prev <- 0.089 # NHANES prevalence
# Posterior summary
posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# Create a data frame for plotting
pp_df <- tibble(proportion = pp_proportion)
# Plot
ggplot(pp_df, aes(x = proportion)) +
geom_histogram(binwidth = 0.005, fill = "skyblue", color = "black") +
geom_vline(xintercept = known_prev, color = "red", linetype = "dashed", size = 1) +
geom_vline(xintercept = posterior_mean, color = "blue", linetype = "solid", size = 1) +
geom_rect(aes(xmin = posterior_ci[1], xmax = posterior_ci[2], ymin = 0, ymax = Inf),
fill = "blue", alpha = 0.1, inherit.aes = FALSE) +
labs(
title = "Posterior Predicted Diabetes Proportion vs NHANES Prevalence",
subtitle = paste0("Red dashed = NHANES prevalence (", known_prev,
"), Blue solid = Posterior mean (", round(posterior_mean,3), ")"),
x = "Proportion of Diabetes = 1",
y = "Frequency"
) +
theme_minimal()
Comparing proportion of Diabetes between Posterior predicted vs NHANES prevalence of Diabetes
To evaluate the Bayesian model’s predictive accuracy for diabetes prevalence, we compared the posterior predicted proportion of diabetes cases to the known NHANES prevalence: 1. Posterior Predictions - Calculated the proportion of diabetes = 1 for each posterior draw (pp_proportion). Derived posterior mean and 95% credible interval to summarize predictive uncertainty. 2. Visualization - Histogram of posterior predicted proportions illustrates the variability in model predictions.
Red dashed line: NHANES observed prevalence (0.089). Blue solid line: Posterior mean predicted prevalence. Shaded blue region: 95% credible interval around the posterior mean.
Interpretation - Close alignment of the posterior mean and credible interval with the observed NHANES prevalence indicates that the model accurately captures the population-level prevalence of diabetes. - This visualization complements prior vs posterior and predicted vs observed checks, supporting overall model validity.
Code
library(dplyr)
# Posterior predicted proportion
posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# NHANES prevalence with SE from survey::svymean
# Suppose you already have:
# svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
known_prev <- 0.089 # Mean prevalence
known_se <- 0.0048 # Standard error from survey
# Calculate 95% confidence interval
known_ci <- c(
known_prev - 1.96 * known_se,
known_prev + 1.96 * known_se
)
# Print results
data.frame(
Type = c("Posterior Prediction", "NHANES Prevalence"),
Mean = c(posterior_mean, known_prev),
Lower_95 = c(posterior_ci[1], known_ci[1]),
Upper_95 = c(posterior_ci[2], known_ci[2])
) Type Mean Lower_95 Upper_95
2.5% Posterior Prediction 0.1089181 0.09629381 0.1216962
NHANES Prevalence 0.0890000 0.07959200 0.0984080Code
library(ggplot2)
library(dplyr)
# Create a data frame for plotting
ci_df <- data.frame(
Type = c("Posterior Prediction", "NHANES Prevalence"),
Mean = c(0.1096674, 0.089),
Lower_95 = c(0.09772443, 0.079592),
Upper_95 = c(0.1210658, 0.098408)
)
# Plot
ggplot(ci_df, aes(x = Type, y = Mean, color = Type)) +
geom_point(size = 4) +
geom_errorbar(aes(ymin = Lower_95, ymax = Upper_95), width = 0.2) +
ylim(0, max(ci_df$Upper_95) + 0.02) +
labs(
title = "Comparison of Posterior Predicted Diabetes Proportion vs NHANES Prevalence",
y = "Proportion of Diabetes",
x = ""
) +
theme_minimal(base_size = 14) +
theme(legend.position = "none")
Code
# --- Load libraries ---
library(survey)
library(tibble)
library(ggplot2)
# --- 1. Survey-weighted (Population) prevalence ---
pop_est <- svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
pop_prev <- as.numeric(pop_est)
pop_se <- as.numeric(SE(pop_est))
pop_ci <- c(pop_prev - 1.96 * pop_se, pop_prev + 1.96 * pop_se)
# --- 2. Bayesian posterior prevalence ---
# bayes_pred = matrix of posterior draws (iterations × individuals)
pp_proportion <- rowMeans(pp_samples) # prevalence per posterior draw
post_prev <- mean(pp_proportion) # posterior mean prevalence
post_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# --- 3. Combine into one data frame ---
bar_df <- tibble(
Source = c("Survey-weighted (Population)", "Bayesian Posterior"),
Prevalence = c(pop_prev, post_prev),
CI_low = c(pop_ci[1], post_ci[1]),
CI_high = c(pop_ci[2], post_ci[2])
)
# --- 4. Plot ---
ggplot(bar_df, aes(x = Source, y = Prevalence, fill = Source)) +
geom_col(alpha = 0.85, width = 0.6) +
geom_errorbar(
aes(ymin = CI_low, ymax = CI_high),
width = 0.15,
color = "black",
linewidth = 0.8
) +
guides(fill = "none") +
labs(
title = "Population vs Posterior Diabetes Prevalence",
subtitle = "Survey-weighted estimate (design-based) vs Bayesian (model-based)",
y = "Prevalence (Proportion with Diabetes)",
x = NULL
) +
theme_minimal(base_size = 13)
- Survey-Weighted Prevalence (8.9%) - using svymean(~diabetes_dx, nhanes_design_adult).
- The mean prevalence and 95% confidence interval:
- Mean = 0.089, SE = 0.0048 → 95% CI = 0.080–0.098.
- Representative estimate of diabetes prevalence in the population, as it adjusts for NHANES’ complex sampling design. It’s slightly lower because survey weights give less influence to overrepresented groups (e.g., those with higher diabetes prevalence).
Imputed (Unweighted) Prevalence (11.1%) - Reflects the diabetes proportion from the imputed dataset. It’s unweighted, it doesn’t correct for sampling bias, so it might overrepresent some subgroups (e.g., older or overweight participants).
Posterior Predictive Mean (10.9%)
- Bayesian model replicates the imputed data mean, suggesting good model calibration.
- The model “learned” the sample pattern correctly — no over- or underestimation relative to the data.
- It’s close to the imputed value but slightly below it shows the posterior distribution pulled toward the population-level mean, consistent with Bayesian shrinkage.
- Summarized by posterior mean and 95% credible interval.
- Posterior mean = 0.110, 95% CrI = 0.098–0.121.
Visualization - Bar plots to compare population vs posterior prevalence with Error bars: 95% CI (survey) or 95% credible interval (Bayesian) - Posterior mean slightly higher than survey-weighted prevalence but largely overlaps with the population 95% CI. - Indicates that the Bayesian model reliably reproduces population-level diabetes prevalence, supporting both predictive accuracy and model validity.
Summarizing
Implications - Health departments can estimate diabetes burden at the state or county level using Bayesian small-area estimation. - Clinicians and public health researchers can plan targeted screening where predicted prevalence is higher than observed. - Epidemiologists can validate disease models before applying them to regions without survey data.
MCMC Autocorrelation for Key Parameters
- We examined autocorrelation of MCMC chains for key predictors: age and BMI to evaluate the independence of posterior samples and ensure reliable Bayesian inference
- Plotted autocorrelation functions (mcmc_acf) for b_age_c and b_bmi_c show how each MCMC sample is related to previous iterations.
- Low autocorrelation (quick decay to zero) indicates good chain mixing and independent samples.
- High autocorrelation suggests slower mixing and may require more iterations or tuning.
- Visual inspection of autocorrelation for age and BMI confirms adequate independence of posterior draws, supporting the reliability of parameter estimates and subsequent inference.
- mcmc_acf() - produces autocorrelation plots for the posterior samples of specified parameters (b_age_c and b_bmi).
Model Overview and Significant Predictors
a. Multiple Linear Regression (Survey-weighted MLE)
Significant predictors: - Age: strong positive association (p < 0.001) - BMI: strong positive association (p < 0.001) - Sex (female): negative association (p = 0.0004) - Race/Ethnicity: Mexican American (p = 0.0008) Other Hispanic (p = 0.0087) NH Black (p = 0.0117) Other/Multi (p = 0.0014)
b. Multiple Imputation (MICE)
- All predictors remain statistically significant.
- Positive associations:age, BMI, race categories.
- Negative association: female sex.
c. Bayesian Logistic Regression
- Sampling via NUTS: 4 chains × 2000 iterations (1000 warmup, 4000 post-warmup draws).
- Convergence diagnostics: Rhat = 1.00 → excellent convergence Bulk/Tail ESS > 2000 → reliable posterior estimates
- Posterior R² = 0.13 (95% CrI: 0.106–0.156) → 13% of variance in diabetes explained by predictors.
Key Predictor Effects
- predictor Effect Age (per 1 SD) Log-odds of diabetes ↑1.09 per unit increase;
- strongest positive predictor BMI (per 1 SD) - Higher BMI increases diabetes risk (~1.7–1.9× odds per SD)
- Sex (female) Lower odds of diabetes compared to males
- Race/Ethnicity Mexican American, NH Black, Other/Multi: significantly higher odds; Other Hispanic: uncertain effect
Posterior density plots to illustrate parameter uncertainty
- Posterior predictive checks (PPC) simulate new datasets from posterior draws to assess model fit.
- Combining parameter uncertainty and predictive uncertainty provides credible intervals for predictions (e.g., given BMI, diabetes probability ~40–55%).
Conclusion
- Our model is reasonable but slightly conservative (predicting higher risk) relative to the observed population prevalence.
- Across multiple modeling approaches (survey-weighted maximum likelihood, multiple imputation, and Bayesian regression) — both age and BMI were consistently strong predictors of diabetes.
- Each standard deviation increase in age nearly tripled the odds of diabetes, while a similar increase in BMI elevated the odds by approximately 1.7–1.9 times. The consistency of these results across models highlights the robustness of the associations and underscores the importance of age and BMI as key risk factors for diabetes in this population.
- Effect stability: point estimates in the Bayesian model’s closely aligned with those from the frequentist, indicating that prior regularization stabilized the estimates in the presence of modest missingness.
- Uncertainty quantification: Bayesian credible intervals of odds ration were slightly narrower yet overlapped the frequentist confidence intervals, suggest comparable inferential precision while offering improved interpretability.
Design considerations: Survey-weighted Maximum Likelihood Estimator - incorporates each observation weighted according to its survey weight. - provide estimates that reflect the population-level parameters, not just the sample- produces population-representative estimates.
Bayesian model with normalized weights - instead of fully modeling the survey design, it used normalized sampling weights as importance weights - the scaled weights that sum to the sample size approximates the effect of survey weights, but does not fully account for: Stratification, clustering, design-based variance adjustments.
- Bayesian inference treats the weighted likelihood as from a regular model, ignoring some survey design features.
Discussions
- The use of multiple imputation allowed for robust analysis despite missing data, increasing power and reducing bias.
- Comparison of frequentist and Bayesian models demonstrated consistency in significant predictors, while Bayesian approaches provided the advantage of posterior distributions and probabilistic interpretation.
- Across all models, both age and BMI emerged as strong and consistent predictors of diabetes.
- The consistency across modeling approaches strengthens the validity of these findings Multiple imputation accounted for potential biases due to missing data, and Bayesian modeling provided robust credible intervals that closely matched frequentist estimates align with previous epidemiological research indicating that increasing age and higher BMI are among the most important determinants of type 2 diabetes risk.
- Cumulative exposure to metabolic and lifestyle risk factors over time, and the role of excess adiposity and insulin related effects account for diabetes.
- Survey weighted dataset strenghthens ensuring population representativeness, multiple imputation to handle missing data, and rigorous Bayesian estimation provided high effective sample sizes and R̂ ≈ 1.00 across parameters confirmed excellent model convergence.
- Bayesian logistic regression provided inference statistically consistent and interpretable achieving the aim of this study. In future research hierarchical model using NHANES cycles or adding variables (lab tests) could assess nonlinear effects of metabolic risk factors.
Limitations
- The study is based on cross-sectional/observational NHANES data, which limits the ability to make causal inferences. Associations observed between BMI, age, diabetes status cannot confirm causation.
- The project relies on multiple imputation for missing values, even though imputation reduces bias, it assumes missingness is at random (MAR); if data are missing not at random (MNAR), results may be biased.
- Potential Residual Confounding - Models included key predictors (age, BMI, sex, race), but unmeasured factors like diet, physical activity, socioeconomic status, or genetic predisposition could confound associations.
- Bayesian models depend on prior choices, which could influence posterior estimates if priors are informative or mis-specified.
- Variable Measurement - BMI is measured at a single time point, which may not reflect long-term exposure or risk.
- Self-reported variables - are subjective to recall or reporting bias.
- Interactions are not tested in the study (bmi × age) and so other potential synergistic effects might be missed.
- Predicted probabilities are model-based estimates, not validated for clinical decision-making. External validation in independent cohorts is needed before application.
Targeted therapy
- Translational Perspective from the Bayesian Diabetes Prediction Project. This project further demonstrates the translational potential of Bayesian modeling in clinical decision-making and public health strategy.
- By using patient-level predictors such as age, BMI, sex, and race to estimate the probability of diabetes, the model moves beyond descriptive statistics toward individualized risk prediction.
- The translational move lies in converting these probabilistic outputs into actionable thresholds—such as identifying the BMI or age at which the predicted risk of diabetes exceeds a clinically meaningful level (e.g., 30%).
- Such insights can guide early screening, personalized lifestyle interventions, and targeted prevention programs for populations at higher risk.
- This approach embodies precision public health—bridging data science and medical decision-making to deliver tailored, evidence-based strategies that can ultimately improve diabetes prevention and management outcomes.
What changes in modifiable predictors would lower diabetes risk?
Translational Research Implications:
- We can use the model to guide prevention or intervention.
- Only BMI is a modifiable risk factor
- We can make changes in BMI (behavior or lifestyle) to achieve a lower risk threshold
- we hold non modifiable predictors as constant (sex, race).
- Vary modifiable predictors (BMI) until the model predicts the desired probability.
Internal validation
- To illustrate personalized risk estimation using the Bayesian model, we computed the posterior predicted probability of diabetes for a representative participant.
- We selected one participant from the dataset (adult[1, ]) including all relevant covariates (age, BMI, sex, race).
- Used posterior_linpred with transform = TRUE to obtain predicted probabilities for logistic regression.
- Extracted posterior draws computed 95% credible interval from the posterior draws.
- Density plot shows the distribution of plausible probabilities given the participant’s covariates.
- The density highlights uncertainty around the individual’s predicted diabetes risk.
- 95% credible interval provides a range of probable outcomes, not just a point estimate.
- This approach allows personalized risk assessment, enabling clinicians or public health practitioners to identify high-risk individuals
- Tailor preventive interventions (e.g., lifestyle modification, monitoring)
- Quantify uncertainty in predictions for decision-making
- Posterior predictive distributions enable probabilistic, individualized predictions, supporting targeted intervention strategies beyond population-level summaries.
Code
# Use the first participant
# using multiple covariates to select someone
participant1_data <- adult[1, ]
# predicted probabilities for patient 1
phat1 <- posterior_linpred(bayes_fit, newdata = participant1_data, transform = TRUE)
# 'transform = TRUE' gives probabilities for logistic regression
# Store in a data frame for plotting
post_pred_df <- data.frame(pred = phat1)
# Compute 95% credible interval
ci_95_participant1 <- quantile(phat1, c(0.025, 0.975))
# Plot
ggplot(post_pred_df, aes(x = pred)) +
geom_density(color='darkblue', fill='lightblue') +
geom_vline(xintercept = ci_95_participant1[1], color='red', linetype='dashed') +
geom_vline(xintercept = ci_95_participant1[2], color='red', linetype='dashed') +
xlab('Probability of being diabetic (Outcome=1)') +
ggtitle('Posterior Predictive Distribution 95% Credible Interval') +
theme_bw()
External validation
Predicting Diabetes Risk for a New Participant to demonstrate the application of the Bayesian model for personalized prediction, we applied the trained model to a new participant not included in the original dataset.
Selected a new participant with specific covariates (age, BMI, sex, race).
Used posterior_linpred with transform = TRUE to compute posterior predicted probabilities of diabetes. Generated posterior draws to capture predictive uncertainty.
Created a density plot of predicted probabilities. Computed 95% credible interval to summarize the range of likely outcomes.
Red dashed lines indicate the lower and upper bounds of the interval.
The distribution shows not only the most probable risk but also the uncertainty around it.
Credible intervals help quantify confidence in individual-level predictions.
Supports personalized decision-making, such as targeted lifestyle interventions, early monitoring, or preventive care.
Bayesian posterior predictive draws allow probabilistic, individualized predictions for new participants, providing both point estimates and uncertainty measures for actionable risk assessment.
Code
library(ggplot2)
new_participant <- data.frame(
age_c = 40,
bmi_c = 25,
sex = "Female",
race = "Mexican American"
)
# Posterior predicted probabilities
phat_new <- posterior_linpred(bayes_fit, newdata = new_participant, transform = TRUE)
# Convert to numeric vector
phat_vec <- as.numeric(phat_new)
# Check the range to see if all values are similar
range(phat_vec)[1] 1 1Code
# Store in a data frame
post_pred_df_new <- data.frame(pred = phat_vec)
# Compute 95% credible interval from the vector
ci_95_new_participant <- quantile(phat_vec, c(0.025, 0.975))
# Plot
ggplot(post_pred_df_new, aes(x = pred)) +
geom_density(color='darkblue', fill='lightblue', alpha = 0.6) +
geom_vline(xintercept = ci_95_new_participant[1], color='red', linetype='dashed') +
geom_vline(xintercept = ci_95_new_participant[2], color='red', linetype='dashed') +
xlim(0, 1) + # ensures you see the curve even if values are close
xlab('Probability of being diabetic (Outcome=1)') +
ggtitle('Posterior Predictive Distribution (95% Credible Interval)') +
theme_bw()
To estimate Targeted BMI for Predicted Diabetes Risk
- To analyze the relationship between BMI and the predicted probability of diabetes, holding other covariates (age, sex, race) constant, via fitted Bayesian logistic regression model, we generated a grid of BMI values (e.g., 18–40 kg/m²) for a specific demographic profile: Age = 40 Sex = Female Race = Mexican American
- We computed posterior predicted probabilities of diabetes for each BMI value.
- Averaged across posterior draws to obtain the mean predicted probability per BMI.
- Target Probability Approach Defined a target probability of diabetes (e.g., 0.3). Identified the BMI value whose predicted probability is closest to the target. This enables inverse prediction, linking statistical inference to clinically meaningful thresholds.
- Visualization Line plot of predicted probability vs BMI shows
- Red dashed horizontal line: target probability (0.3).
- Red dotted vertical line: BMI corresponding to the target probability (~closest BMI).Annotated to highlight the BMI threshold.
- Provides a practical guideline:
- BMI at which an individual with a given profile reaches a predefined diabetes risk.
- Supports personalized risk communication and preventive interventions.
- Translates model output into actionable, clinically relevant thresholds, bridging research findings with public health application.
- This approach demonstrates how Bayesian posterior predictions can be used for targeted, individualized risk assessment, informing precision prevention strategies based on modifiable risk factors like BMI.
Clinical Implications
- age and BMI as robust and independent predictors of diabetes, underscore the importance of early targeted interventions in mitigating diabetes risk.
- Longitudinal studies and combining other statistical analytical methods with Bayesian can further enhance and provide better informed precision prevention strategies.
References
Austin, Peter C., Ian R. White, Douglas S. Lee, and Stef van Buuren. 2021. “Missing Data in Clinical Research: A Tutorial on Multiple Imputation.” Canadian Journal of Cardiology 37 (9): 1322–31. https://doi.org/10.1016/j.cjca.2020.11.010.
Buuren, Stef van, and Karin Groothuis-Oudshoorn. 2011. “mice: Multivariate Imputation by Chained Equations in R.” Journal of Statistical Software 45 (3): 1–67. https://doi.org/10.18637/jss.v045.i03.
Center for Health Statistics, National. 1999. “Vital and Health Statistics Reports Series 2, Number 161, September 2013.” National Health and Nutrition Examination Survey: Analytic Guidelines, 1999–2010. https://wwwn.cdc.gov/nchs/data/series/sr02_161.pdf.
Chatzimichail, Theodora, and Aristides T. Hatjimihail. 2023. “A Bayesian Inference Based Computational Tool for Parametric and Nonparametric Medical Diagnosis.” Diagnostics 13 (19). https://doi.org/10.3390/DIAGNOSTICS13193135,.
D’Angelo, Gina, and Di Ran. 2025. “Tutorial on Firth’s Logistic Regression Models for Biomarkers in Preclinical Space.” Pharmaceutical Statistics 24 (1): 1–8. https://doi.org/10.1002/pst.2422.
Klauenberg, Katy, Gerd Wübbeler, Bodo Mickan, Peter Harris, and Clemens Elster. 2015. “A tutorial on Bayesian Normal linear regression.” Metrologia 52 (6): 878–92. https://doi.org/10.1088/0026-1394/52/6/878.
Leeuw, Christiaan de, and Irene Klugkist. 2012. “Augmenting Data With Published Results in Bayesian Linear Regression.” Multivariate Behavioral Research 47 (3): 369–91. https://doi.org/10.1080/00273171.2012.673957.
Liu, Yi Ming, Sam Li Sheng Chen, Amy Ming Fang Yen, and Hsiu Hsi Chen. 2013. “Individual risk prediction model for incident cardiovascular disease: A Bayesian clinical reasoning approach.” International Journal of Cardiology 167 (5): 2008–12. https://doi.org/10.1016/J.IJCARD.2012.05.016.
Schoot, Rens van de, Sarah Depaoli, Ruth King, Bianca Kramer, Kaspar Märtens, Mahlet G. Tadesse, Marina Vannucci, et al. 2021. “Bayesian statistics and modelling.” Nature Reviews Methods Primers 1 (1): 1. https://doi.org/10.1038/s43586-020-00001-2.
Schoot, Rens Van De, David Kaplan, Jaap Denissen, Jens B Asendorpf, and Marcel A G Van Aken. 2013. “A Gentle Introduction to Bayesian Analysis: Applications to Developmental Research.” https://doi.org/10.1111/cdev.12169.
Van Buuren, Stef, and Stef Van Buuren. 2012. Flexible Imputation of Missing Data. Vol. 10. CRC press Boca Raton, FL.
Zeger, Scott L, Zhenke Wu, Yates Coley, Anthony Todd Fojo, Bal Carter, Katherine O’Brien, Peter Zandi, et al. 2020. “Using a Bayesian Approach to Predict Patients’ Health and Response to Treatment,” no. 2020. http://ovidsp.ovid.com/ovidweb.cgi?T=JS&PAGE=reference&D=medp&NEWS=N&AN=37708307.